Solving an One-Dimensional Cutting Stock Problem by Simulated Annealing and Tabu Search

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Jahromi, Meghdad HMA; Tavakkoli-Moghaddam, Reza; Makui, Ahmad; Shamsi, Abbas

Article Solving an one-dimensional cutting stock problem by simulated annealing and tabu search

Journal of Industrial Engineering International

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Suggested Citation: Jahromi, Meghdad HMA; Tavakkoli-Moghaddam, Reza; Makui, Ahmad; Shamsi, Abbas (2012) : Solving an one-dimensional cutting stock problem by simulated annealing and tabu search, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 8, pp. 1-8, http://dx.doi.org/10.1186/2251-712X-8-24

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ORIGINAL RESEARCH Open Access Solving an one-dimensional cutting stock problem by simulated annealing and tabu search Meghdad HMA Jahromi1*, Reza Tavakkoli-Moghaddam2, Ahmad Makui3 and Abbas Shamsi1

Abstract A cutting stock problem is one of the main and classical problems in operations research that is modeled as LP problem. Because of its NP-hard nature, finding an optimal solution in reasonable time is extremely difficult and at least non-economical. In this paper, two meta-heuristic algorithms, namely simulated annealing (SA) and tabu search (TS), are proposed and developed for this type of the complex and large-sized problem. To evaluate the efficiency of these proposed approaches, several problems are solved using SA and TS, and then the related results are compared. The results show that the proposed SA gives good results in terms of objective function values rather than TS. Keywords: One-dimensional cutting stock problem, Mathematical model, Simulated annealing, Tabu search

Background All stock lengths are identical and there is no A one-dimensional cutting stock problem (1D-CSP) is one difference between the lengths of them. of the famous combinatorial optimization problems, which has many applications in industries. In this prob- In a standard cutting problem, each cutting stock is lem, the amount of residual pieces of processed stock rolls, consumed totally, because the remaining is as cutting called trim loss (i.e., wastage) is a significant objective in waste. While in a general one-dimensional stock cutting most studies that should be minimized. A standard one- problem (G1D-CSP), as another kind of the 1D-CSP, dimensional cutting stock problem (S1D-CSP) as a kind stock cutting residuals can be used later. Also, in a GlD- of the above problem is known as an NP-complete one CSP, the storage stock has different lengths. Reduction (Gradisar et al. 2002). Finding solutions for this problem es- of cutting wastes is one of the main goals in the cutting pecially for large-sized problems is extremely difficult. In process and also one of the basic purposes in the 1D- fact, due to the NP-hardness of this problem, it is computa- CSP. A standard one dimensional cutting stock problem tionally intractable to obtain its optimum solution in a rea- (S1D-CSP) and different procedures have been reviewed sonable time for medium to large-sized problems, and by many researchers. (Varela et al. 2009) worked on a hence researchers have tried to develop meta-heuristic practical cutting stock problem from a production plant algorithms for solving such a problem. In fact, applying of plastic rolls as a variant of the well-known one dimen- these algorithms is not easy to reliably state that which of sional cutting stock. They proposed a heuristic solution these algorithms is best for the given problem. However, it based on a GRASP algorithm in their work. is possible to identify methods that consistently produce (Cui 2005) proposed an algorithm to produce T-shape better results compared to others for a certain problem. cutting patterns applied in the manufacturing of circular TheassumptionsoftheS1D-CSPareasfollows. and sectional blanks for stators and rotors. The proposed algorithm uses the knapsack algorithm and an In this problem, all used stock lengths should be cut enumeration method to determine the optimal combin- to the end in as much as it is possible. ation of blank rows in the strips, the strip numbers and directions in the pattern. A heuristic algorithm for the one-dimensional cutting stock problem with usable left- * Correspondence: [email protected] 1Department of Industrial Engineering, Khomein Branch, Islamic Azad over is presented by (Cui & Yang 2010). University, Khomein, Iran Full list of author information is available at the end of the article

© 2012 Jahromi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 2 of 8 http://www.jiei-tsb.com/content/8/1/24

Their algorithm consists of two procedures, namely with the minimum weighted tardiness. (Liang et al. linear programming and sequential heuristic procedures 2002b) applied the newest perfect algorithm, which is that fulfill the major and remaining portions of the item similar to genetic algorithm, with a mobility function demands, respectively. This algorithm is able to balance for a cutting stock problem. (Weng et al. 2003) ap- the cost of the consumed bars, the profit from left- plied the GA to the manufacturing rule oriented cut- overs and the profit from a shorter stocks reduction. ting of a sectional steel problem for the shipbuilding (Armbruster 2002) presented a solution approach for industry. the capacitated one-dimensional cutting stock problem (Sung et al. 2004) applied tabu search (TS) to develop under technological constraints by concentrating on an optimization procedure based on the cutting pro- solving the pattern sequencing problem occurring from cesses in the shipbuilding industry. (Yang et al. 2006) the technological constraints. (Dikili et al. 2007) devel- developed the efficient TS method with a mixed object- oped a novel approach for one-dimensional cutting stock ive function for the one dimensional cutting stock prob- problems in ship production via the cutting patterns lem under a small amount of various cutting conditions. obtained by the analytical methods at the mathematical Their proposed TS method consists of a mixed objec- modeling stage. The proposed method is able to cap- tive function that helps the algorithm to discover the ture the ideal solution of analytical methods. The main best solution during the one-by-one neighborhood advantage of their proposed method is to guarantee in searching processes. (Gradisar et al. 2002) presented obtaining an integer solution. Also, the method behaves an experimental study of various methods aimed at so that the trim loss is minimized and the stock usage minimizing trim-loss, in their work they mentioned is maximized. that the LP-based method is possible only when the (Liang et al. 2002a) proposed an evolutionary algo- stock is of the same length or few groups of stand- rithm based on evolutionary programming for cutting ard lengths. When all stock lengths are different fre- stock problems considering contiguity. Two new muta- quencies, an approach can be used that treat by each tion operators are proposed in their work. Results item individually. found by the evolutionary programming are signifi- In addition, (Chen et al. 1996) proposed a simulated cantly better than or comparable to those found by annealing (SA) method for the one-dimensional cutting the genetic algorithm (GA). It is worth to note that stock problem. A meta-heuristic method based on ACO evolutionary programming uses mutation as the pri- was presented to solve a one-dimensional cutting stock mary search operator and does not employ any problem by (Eshghi & Javanshir 2005). In their work, crossover. based on probabilistic laws that designed, ants do select (Poldi & Arenales 2009) proposed heuristics for the various cuts and then select the best patterns. Also, one-dimensional cutting stock problem with limited (Dyckhoff 1990) developed a classification scheme for multiple stock lengths. They reviewed some existing cutting stock problems through a large variety of appli- heuristics and proposed others for solving the integer cations reported in the literature so that classified pro- one-dimensional cutting stock problem with multiple blems using four characteristics; namely, dimensionality, stock lengths. Their study deals with the classical one- kind of assignment, assortment of large objects and as- dimensional integer cutting stock problem. The given sortment of small items. He classified the cutting stock objective function is to minimize the waste (i.e., problems solutions into two groups: 1) item-oriented trim loss). approach that treats each item to be cut separately (Umetani et al. 2003) considered the 1D-CSP, in which and 2) pattern-oriented approach. So that in this the number of different cutting patterns is constrained approach at the beginning order lengths are combined within a given bound. They proposed a meta-heuristic into cutting patterns, for which the frequencies that are algorithm incorporated with an adaptive pattern gener- necessary to satisfy the demands, are determined in a ation technique. The proposed meta-heuristic searches a succeeding step. In the second approach, the classic LP- solution with small deviations from the given demands based namely “delayed pattern generation” proposed while using the fixed number of different cutting pat- (Gilmore & Gomory 1961; Gilmore & Gomory 1963) or terns. (Reinertsen & Vossen 2010) considered the prob- any other LP-based method is mostly used. (Javanshir & lem of cutting stock problems with due date while Shadalooee 3) developed a mathematical model for 1D- addressing common cutting considerations, such as CSP by defining the virtual cost for the trim loss of each aggregation of orders, multiple stock lengths and cutting stock. They solved their developed model with SA algo- different types of material on the same machine. In their rithm. Also, 1D-CSP considered in their work is taken work, meeting due dates is more important than redu- into account as item-oriented. cing the scrap. Also, in condition that there is no feas- It is not easy to define exactly which meta-heuristic ible production plan, their models find a cutting plan algorithm is best for which type of problem. However, it Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 3 of 8 http://www.jiei-tsb.com/content/8/1/24

is possible, for a given problem, to identify algorithms Mathematical model that consistently produce better results compared with The MILP model for a standard one-dimensional cutting those results produced by other algorithms. For this rea- stock problem can be defined as follows. son, in this paper, we compare the efficiency and effectiveness of two well-known meta-heuristic, namely simulated Xm annealing (SA) and tabu search (TS), in producing the min T ¼ tlj ð1Þ cutting plan with the lowest trill loss. In addition, we j¼1 present a mixed-integer linear programming (MILP) s.t. mathematical model for a standard one-dimensional cutting stock problem that minimizes the trim losses of the Xm ¼ 8 ð Þ cutting orders. Because of the NP-hard nature of this xij ni i 2 problem, the presented mathematical model can find an j¼1 optimal solution by a standard OR software (e.g., Lingo) Xn for small-sized problems. xij:si þ tlj ¼ dj:yj 8j ð3Þ This paper is organized as follows. Section 2 presents i¼1 the mathematical programming model for the S1D-CSP. y 2ð0; 1Þ Simulated annealing (SA) and tabu search (TS) algo- j rithms for the considered problem are proposed in xi;j 2 integer Sections 3 and 4, respectively. The computational ex- perience is represented in Section 5. Finally, conclusion Minimizing the total trim loss is the objective function is made in Section 6. (1) of the model. Constraint (2) guarantees cutting needs of orders with regards to the storage stock. Constraint (3) calculates the cutting waste of each stock in the cut- Mathematical model of S1D-CSP ting process. This section presents a mixed-integer linear program- Due to NP-hard nature of this problem, the computa- ming (MILP) model for a standard one-dimensional cut- tional time increases exponentially and finding optimal ting stock problem (S1D-CSP). Minimizing the sum of solution for large-size problem is extremely difficult. cutting wastes or cutting remaining unusable in satisfy- This model is solved by the Lingo software using a ing of needs (demands) is the objective function of the branch-and-bound method, and the computational times presented model. and the state of the method for 20 randomly generated problems with different sizes are shown in Table 1. Results show finding optimal solution for large-scale Notations problems (15, 16, 17 and 18), in a reasonable time To formulate the S1D-CSP, the following notations are and gradually with increasing the size of the problems used. (19 and 20), even finding feasible area of solutions, is so hard. i i= ,...,n order number ( 1 ). Since finding the optimal solution in a reasonable time j j= ,...,m stock number ( 1 ). for medium to large-sized problems is intractable, a d j stock length. number of researchers have considered developing tl j stock wastes. meta-heuristics to compute near-optimal solutions. It is s i order lengths. noticeable that identifying methods consistently produ- n s i required number of orders with i length. cing better results, as compared to others, for a certain T sum of the cutting wastes in the cutting problem is very important and interesting. Among the program. meta-heuristics, simulated annealing (SA) and tabu search (TS) are the ones whose applications have Decision variables been widely explored for different combinatorial opti- The following variables are used in the presented MILP mization problems. In this work, effectiveness and effi- model. ciency of these meta-heuristics for the given problem are compared. xi,j integer variable, number of orders with si length that are cut from stock j. Simulated annealing for S1D-CSP yj zero–one variable that equals to one if the stock (Kirkpatrick et al. 1983) introduced simulated annealing j is applied in the cutting plan otherwise, equals (SA) as a meta-heuristic algorithm in 1983. It draws its to zero. inspiration from physical annealing of solids. In this Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 4 of 8 http://www.jiei-tsb.com/content/8/1/24

Table 1 NP-hard nature of the S1D-CSP each temperature, called the Markov chain length. The Problems Requested State CPU neighborhood search structure generates a new solution orders time from the current candidate solution by slightly changing (Sec.) it. How move from one solution to another one in the 1 9 Global optimum 1 solution space is dependent on the structure of design- found 215 1ing operators. Attaining to the undiscovered area of 332 2exploration space pertain to design efficient operators. 445 4The generic flow of SA is given below. 559 7

672 10Select an initial temperature T0 >0; 785 28Generate an initial solution, S0, and make it the current 898 59solution, Scurrent , and the current best solution, Sbest; 9 110 607 repeat 10 190 2570 n 11 305 6077 set repetition counter =1; 12 950 15185 repeat 13 1503 58106 14 1704 79202 Generates the solution Snew in the neighborhood of S 15 1850 Cannot find any global current; optimum after a Δ = f(Scurrent)-f(Snew); 16 1990 Calculate 24-hours run If (Δ ≤ 0) then Scurrent =Snew; 17 2500 else Scurrent =Snew with the probability P(Δ,T); 18 3645 if (Snew Markov chain length;

Reduce the temperature T; algorithm, it may become stuck in a local optimum. until the stopping criterion is met. To prevent this from happening and trapping in a local optimum, the algorithm occasionally makes a random change to the solution even through the Solution representation change does not immediately improve the result. This The first step in developing of a meta-heuristic proced- may allow the program to break free of a local ure is to design the solution representation. In this optimum and find a better solution later. If the change paper, with regards to the problem nature, the initial so- does not lead to a better solution, the algorithm will lution is created randomly. This solution is constructed probably undo the change in a short time anyway. SA as a (1 ×m)-array, so that m equal to the number of consists of internal and external loops. Task of the in- orders. Each order placed in a cell randomly as illu- ternal loop controls the attaining balance in each strated in Figure 1. temperature and updating temperature. The Metropolis decision-making loop (i.e., possibility of acceptance of non-improvement) is inserted in the in- Neighborhood generation (perturbation) ternal loop. According to the decision-making loop, if To search solution space, attaining to undiscovered the current solution Scurrent cannot improve the objec- areas and finding the best solutions, using suitable tive function comparing with the new neighboring solu- neighborhood search operators is required. As stated Δ tion Snew; then, if y < e T (i.e., Δ=E(Snew)-E(Scurrent)>0 before, designing this operator is dependent on the so- and y 2[0,1]), the new solution will be accepted. In fact, lution representation. Mechanism of the neighborhood the acceptance possibility of a neighboring solution is generation should be able to construct the entire solu- dependent on changing the amount of the objective tion space. In this paper, in order to generate the function, and current temperature. The value of T starts neighboring solution, the algorithm randomly changes at initial temperature T0 and is slowly reduce according the place of two orders in the current solution in its to the cooling ratio a <1(i.e., Tk+1 = a.Tk). array, as shown in Figure 2. This strategy is very sim- Also, the external loop controls the stopping condition ple and guarantees searching the whole of the solution of SA. A given SA may iterate for a number of times at space. Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 5 of 8 http://www.jiei-tsb.com/content/8/1/24

inner loop of SA as a counter. The Markov chain length O1 O3 O4 O2 considered in our proposed SA depends on the problem Figure 1 Representation of a solution with four orders. size (i.e., number of orders).

Fitness function Outer counter variable The objective function considered in this paper mini- The outer counter variable is increased by one, when- mizes the total trim loss. It is assumed that the stock ever a solution is not selected by probability and length is standard with the equal length (e.g., 12 meters shows the number of rejection of perturbed solutions. considered in this paper). By considering the position of In fact, if this variable reaches to a pre-determined fixed each order in the array from the first (i.e., leftmost) to value as input parameter, the search procedure is the end (i.e., rightmost), the total waste is computed. So stopped or started again depending on the other criteria. that all the orders, which the summation of their length Because there is no superior solution exists in the neigh- is lesser than (or equal to) the length of stock, are borhood, and the search is reached to a near optimal/ assigned to it for cutting and the difference between optimal solution. them considered as the trill loss of that stock and so on, till all orders are assigned. Stopping criterion In our proposed SA, the stopping criterion is a number Temperature decreasing function of consecutive times in the cooling process that does Temperature is used to compute the acceptance prob- not lead to improvement of the objective function as ability of a solution that is worse than the previous one. input data. In the proposed SA algorithm, a geometric function is used as follows. Tabu search for S1D-CSP (Glover 1990) first proposed tabu search (TS) as a Tk ¼ αTk1 ð4Þ higher-level heuristic algorithm used for solving optimization problems. It avoids entrapment in cycles by where, α is a constant less than 1, but it is very close to forbidding or penalizing moves that take the solution, 1 (i.e., a typical value for α is chosen between 0.85 and in the next iteration, to points in the solution space o.95). k is the temperature transformations counter. The previously visited by using the list of prohibited neigh- high amount of α is as the slow temperature reduction boring solutions known as tabu list. Aspiration defined in that leads to the best search of neighbors; but, the solu- the tabu search restricts the search from being trapped tion time increases. Adversely, less amount of α is an into a solution surrounded by tabu neighbors. Thus, it indicator of the quick temperature reduction and leads helps in the checking condition for the acceptance of to quicker search in neighbors and sooner coverage in solutions, known as a main feature of TS. It also allows the algorithm. search to override the tabu status of the solution and provides backtracking of the recent solutions as they lead Markov chain length to a new path towards a better solution. The generic One of the most important parameters in determining structure of the TS algorithm is given below. the quality of resultant solutions of SA is the number of searched points in solution space at each temperature, Step 1. (initialization) called Markov chain length. In our proposed SA, this (A). Generate initial solution Sinitial counter is denoted by MarChain_Length and used in the (B). Scurrent = Sinitial

O1 O3 O4 O2

O4 O3 O1 O2 Figure 2 Neighborhood generation. Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 6 of 8 http://www.jiei-tsb.com/content/8/1/24

(C). Record the current best known solution by Table 2 Result obtained from SA and TS and branch-and- setting Sbest = Scurrent ,definebest_cost = f(Sbest) bound methods (D). Set tabu list (H) to empty. Problems B&B method SA TS OFV* OFV GAP OFV GAP Step 2. (choice and termination) 1 4.5 4.5 0.0 4.5 0.0 Candidate-N S Determine ( current) as subset of 2 7.5 7.5 0.0 7.5 0.0 S N(H, current). 3 16 16 0.0 16 0.0 Select Snew from Candidate-N(Scurrent). 4 22.5 22.5 0.0 22.5 0.0 (Snew is called a highest evaluation element of 5 29.5 29.5 0.0 29.5 0.0 Candidate-N(Scurrent). Terminate by a chosen iteration cut-off rule. 6 36 38 5.6 40 11.1 Step 3. (update) 7 34 37 8.8 39 14.7 Re-set Scurrent = Snew. 8 39.2 43 9.7 44 12.2 If f(Scurrent)

Table 3 Comparison between SA and TS algorithms Problems Requested SA TS orders Stocks used Trim loss Total trim loss ratio (%) Stocks used Trim loss Total trim loss ratio (%) 1 9 5 4.5 7.50 5 4.5 7.50 2 15 9 7.5 6.94 9 7.5 6.94 3 32 15 16 8.89 15 16 8.89 4 45 25 22.5 7.50 25 22.5 7.50 5 59 34 29.5 7.23 34 29.5 7.23 6 72 42 38 7.54 44 40 7.58 7 85 51 37 6.05 56 39 5.80 8 98 55 43 6.52 61 44 6.01 9 110 60 45 6.25 69 50 6.04 10 190 111 79 5.93 128 85 5.53 11 305 149 125.4 7.01 173 141 6.79 12 950 501 302 5.02 554 336 5.05 13 1503 850 511 5.01 865 540 5.20 14 1704 951 599.5 5.25 1097 612 4.65 15 1850 1150 751 5.44 1165 1170 8.37 16 1990 1110 897 6.73 1222 1412.4 9.63 17 2500 1350 993 6.13 1480 1955.6 11.01 18 3645 1901 1287 5.64 1984 2192.4 9.21 19 4985 2713 2004.2 6.16 2671 3198 9.98 20 5750 3112 2614 7.00 3221 3930.8 10.17 Average 6.49 7.45

Conclusions known meta-heuristics based on simulated annealing In this paper, a mixed-integer linear programming (SA) and tabu search (TS) have been proposed and (MILP) model for a standard one-dimensional cutting developed. To compare the efficiency of these algo- stock problem (S1D-CSP) has been presented. Due to its rithms, a number of test problems have been solved, and NP-hardness, finding feasible solution in a reasonable the associated results in terms of the objective function time, especially for medium to large size problems, is (i.e., trim loss) and CPU time have been compared. The impossible. To solve such a hard problem, two well- results show the efficiency of our proposed SA in finding

TS algorithm SA algorithm 2100

1800

1500

1200

900

600 CPU time(second) 300

0 1 2 3 4 5 6 7 8 9 1011121314151617181920 Problems Figure 3 CPU times of the proposed SA and TS algorithms. Jahromi et al. Journal of Industrial Engineering International 2012, 8:24 Page 8 of 8 http://www.jiei-tsb.com/content/8/1/24

cutting plan as its output with lesser trim loss rather Varela R, Vela CR, Puente J, Sierra M, González-Rodríguez I (2009) An effective than TS. In addition, the results show that TS outper- solution for a real cutting stock problem in manufacturing plastic rolls. Ann Oper Res 166(1):125–146 forms SA in terms of the CPU time. Weng WC, Yang CT, Hung CF (2003) The optimization of section steel arrangement for ship construction associated with cutting rule by genetic Competing interests algorithm. In: Proceeding of the 17th Asian-Pacific Technical Exchange and The authors declare that they have no competing interests. Advisory Meeting on Marine Structure. National Cheng Kung University, Tainan, Taiwan, 237–246 Authors’ contributions Yang CT, Sung TC, Weng WC (2006) An improved tabu search approach with MHMAJ studied the associated literature review and presented the mixed objective function for one-dimensional cutting stock problems. Adv – mathematical model. RTM managed the study and participated in the Eng Softw 37(8):502 513 problem solving approaches. AM approved and validate the model and was responsible for revising the manuscript. AS managed and analyzed the doi:10.1186/2251-712X-8-24 required data and conducted the experimental analysis. All authors read and Cite this article as: Jahromi et al.: Solving an one-dimensional cutting approved the final manuscript. stock problem by simulated annealing and tabu search. Journal of Industrial Engineering International 2012 8:24. Author details 1Department of Industrial Engineering, Khomein Branch, Islamic Azad University, Khomein, Iran. 2Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran. 3Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran.

Received: 1 December 2010 Accepted: 3 March 2012 Published: 4 October 2012

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